Optimally small sumsets in finite abelian groups

Let G be a finite abelian group of order g. We determine, for all 1 r,s g, the minimal size μG(r,s)=minA+B of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μG(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μG are recalled in the Introduction